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Full text of "Treatise On Analysis Vol-Ii"

30       XII   TOPOLOGY AND TOPOLOGICAL ALGEBRA

(c)    Deduce from (a) and (b) that

A ^ I/log 4,       ^ 5 I/log 2

N

(use the fact that £ 1/(N + r) tends to log 2 as N -»• + <x>).

r=l

(d)    Take an = Iog2(2n - 1) - [Iog2(2w - 1)]. Show that we have A = I/log 4 and
p, = i/iog 2 for this sequence. (Observe that, if N = 2P, the set {aif..., <ZN} is the same
as the set {CN ,..., c2N -1} where cn = Iog2 n — [Iog2 «].) (Cf. Section 13.21, Problem 15.)

7.    Let E be a metrizable space. Show that a nonempty closed subset S of E is the support
of a lower semicontinuous (resp. continuous) real-valued function ^0 on E if and only
if S is the closure of its interior.

8.    Let E be a metrizable space and / a mapping of E into ft. The lower semicontinuous
regularization of/is defined to be the upper envelope of the continuous functions g on
E such that g<*f.

(a)    For each x e E we denote by lim inf f(y) the greatest lower bound of the numbers

y-*x

lim inf/(>>„), for all sequences (yn) tending to x Show that the function ;eh->-lim inf/0>)

n-»oo                                                                                                                                                             y-*x

is the lower semicontinuous regularization of/.

(b)    Suppose that E is an open set in R. For each x e E, let lim inf f(y) denote the

y-»x, y£x
greatest upper bound of the numbers lim inf/OB) for all sequences (yn) such that

n-*oo

yn -»• x and yn ^> x for all n. Show that the set of all x e E such that
lim inf f(y) > lim inff(y)

y-*x, y&x                   y-+x

is at most denumerable. (For each pair (p, q) of rational numbers with p>g, show
that the set of points x e E such that

lim inf f(y) >p>q>\im Mf(y)
y-+xt y&x                                     y-*x

is at most denumerable, by following the method of Section 3.9, Problem 3.)

9.    A Dirichlet series is a series whose general term is of the form ane~*n5, where (An)
is an increasing sequence of real numbers, tending to + oo, and (an) is any sequence
of complex numbers, and s is complex.

(a) Show that if the series is convergent for 5- = s0, then it is uniformly convergent in
the angular sector consisting of the points s = s0 + pew with p ^> 0 and — a £ 6 <i a,
where a is such that 0 < a < 7r/2. (Reduce to the case s0 — 0; show first that if
9ts = or, then

\e-**^e~**\ g |jio-H*""-«-**)

Jb
e~~xs dx. For each
a

integer m and each n^m, put S™, „ — am + am+i H-------h an and use the identity

(AbeFs partial summation formula)reater than or equal